movingmesh

Moving mesh, or dynamic/adaptive mesh, is a numerical technique used to discretize a spatial domain where the mesh itself evolves over time. The primary aim is to concentrate computational elements in regions requiring higher resolution, such as steep gradients, shocks, interfaces, or localized features, while reducing resolution in smoother areas. This results in improved accuracy without a proportional increase in the total number of elements.

Principles and methods

Moving mesh relies on defining a rule for how mesh nodes or elements should migrate as the solution evolves. Common approaches include equidistribution principles using a monitor function that signals where resolution is needed, and metric-based formulations that specify desired element density and shape in a given region. The movement can be governed by partial differential equations, often referred to as moving mesh PDEs (MMPDEs), or by variational principles that minimize a mesh-energy functional. In fluid and continuum simulations, the mesh movement is typically integrated with the physical model in one of several frameworks: Lagrangian motion, where the mesh follows material flow; Eulerian or fixed mesh with adaptation; and Arbitrary Lagrangian-Eulerian (ALE), which decouples mesh motion from physical motion to control distortion and stability.

Implementation and challenges

In practice, moving mesh involves computing a mesh velocity or updating a mesh map, interpolating solution data between meshes, and sometimes remeshing when element quality degrades. Remeshing can preserve accuracy but introduces interpolation errors and potential conservation issues. Key challenges include maintaining mesh quality to avoid tangled or distorted elements, ensuring numerical stability and conservation properties across mesh changes, and efficiently solving the coupled mesh and physical equations.

Applications

Moving mesh is used in computational fluid dynamics, aerodynamics, astrophysical simulations, climate and weather models, fracture and phase-field problems, and biomechanics, wherever localized phenomena necessitate adaptive spatial resolution.

See also

Adaptive mesh refinement, ALE methods, mesh optimization.